Definition:Harmonic Wave
Definition
A harmonic wave is a wave propagated without change of shape whose wave profile can be expressed as a sine curve.
Hence as a sine curve is a periodic real function, the following definitions also all apply to harmonic waves:
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Amplitude
Let $\phi$ be a harmonic wave expressed as:
- $\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The amplitude of $\phi$ is the constant $a$.
Wavelength
Let $\phi$ be a harmonic wave expressed as:
- $\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The wavelength $\lambda$ of $\phi$ is the period of the wave profile of $\phi$.
Period
Let $\phi$ be a harmonic wave expressed as:
- $\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The period $\tau$ of $\phi$ is the time taken for one complete wavelength of $\phi$ to pass an arbitrary point on the $x$-axis.
Frequency
Let $\phi$ be a harmonic wave expressed as:
- $\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The frequency $\nu$ of $\phi$ is the number of complete wavelengths of $\phi$ to pass an arbitrary point in unit time.
Wave Number
Let $\phi$ be a harmonic wave expressed as:
- $\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The wave number $k$ of $\phi$ is the number of complete wavelengths of $\phi$ per unit distance along the $x$-axis.
Phase
Let $\phi_1$ and $\phi_2$ be a harmonic waves expressed in wave number and frequency form as:
| \(\ds \forall x, t \in \R: \, \) | \(\ds \map {\phi_1} {x, t}\) | \(=\) | \(\ds a \map \cos {2 \pi \paren {k x - \nu t} }\) | |||||||||||
| \(\ds \map {\phi_2} {x, t}\) | \(=\) | \(\ds a \map \cos {2 \pi \paren {k x - \nu t} + \epsilon}\) |
The phase of $\phi_1$ relative to $\phi_2$ is the quantity $\epsilon$.
Also see
- Results about harmonic waves can be found here.
Sources
- 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$